816 CHAPTER 12 Statistics 100 130 0 2.00 Original values z-scores 97.72% Figure 12.31 100 70 Original values 0 –2.00 z-scores 2.28% Figure 12.32 130 100 0 115 1 2 Original values z-scores 13.59% Figure 12.34 The percent of individuals with IQ scores below 115 is the same as the percent of data below a z-score of 1.00 (Fig. 12.30(b)). Since our z-score is positive, we use Table 12.8(b). From Table 12.8(b), we determine that the area to the left of a z-score of 1.00 is .8413. Therefore, 84.13% of all the IQ scores are below a z-score of 1.00. Thus, 84.13% of individuals have IQ scores below 115. b) Begin by determining the z-score for 130. = − = = z 130 100 15 30 15 2.00 130 The percent of data below an IQ score of 130 is the same as the percent of data below a z-score of 2.00 (Fig. 12.31). Using Table 12.8(b), we determine that the area to the left of a z-score of 2.00 is .9772. Therefore, 97.72% of the IQ scores are below a z-score of 2.00. Thus, 97.72% of all individuals have IQ scores below 130. c) Begin by determining the z-score for 70. = − = − = − z 70 100 15 30 15 2.00 70 The percent of data below an IQ score of 70 is the same as the percent of data below a z-score of 2.00 − (Fig. 12.32). Since the z-score is negative, we use Table 12.8(a). Using the table, we determine that the area to the left of = − z 2.00 is .0228. Therefore, 2.28% of the data are below a z-score of −2.00. Thus, 2.28% of all individuals have IQ scores below 70. d) In part (a), we determined that = z 1.00, 115 and in part (c), we determined that = − z 2.00. 70 The percent of data below a z-score of 1.00 is 84.13% (Fig. 12.33(a)). The percent of data below a z-score of −2.00 is 2.28% (Fig. 12.33(b)). Since we want to determine the percent of data between two z-scores, we subtract the smaller percent from the larger percent: − = 84.13% 2.28% 81.85% (Fig. 12.33(c)). Thus, 81.85% of all individuals have IQ scores between 70 and 115. 100 0 70 –2.00 1.00 (a) 115 1.00 115 1.00 115 Original values – = z-scores 100 0 70 –2.00 (b) Original values z-scores 100 0 70 –2.00 (c) Original values z-scores 2.28% 84.13% 81.85% Figure 12.33 e) In part (a), we determined that = z 1.00, 115 and in part (b), we determined = z 2.00. 130 The percent of data below a z-score of 1.00 is 84.13%. The percent of data below a z-score of 2.00 is 97.72%. Since we want to determine the percent of data between two z-scores, we subtract the smaller percent from the larger percent: − = 97.72% 84.13% 13.59% (Fig. 12.34). Thus, 13.59% of all individuals have IQ scores between 115 and 130. f) Begin by determining a z-score for 122.5. = − = = z 122.5 100 15 22.5 15 1.50 122.5 The percent of IQ scores above 122.5 is the same as the percent of data above = z 1.50 (Fig. 12.35). To determine the percent of data above = z 1.50, we can determine the percent of data below = z 1.50 and subtract this percent from 100%. In Table 12.8(b), we see that the area to the left of = z 1.50 is .9332. Therefore, 93.32% of the IQ scores are below = z 1.50. The percent of IQ scores above = − z 1.50 are100% 93.32%, or 6.68%. Thus, 6.68% of all IQ scores are greater than 122.5. 7 Now try Exercise 49 100 0 122.5 1.5 Original values z-scores 6.68% 93.32% Figure 12.35 Instructor Resources for Section 12.5 in MyLab Math • Objective-Level Videos 12.5 • Interactive Concept Video: Understanding z-scores • Animation: Exploring z-scores • PowerPoint Lecture Slides 12.5 • MyLab Exercises and Assignments 12.5
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